Category A COMPANION TO Theoretical Econometrics

Even when a vector of time series is I(1), the size of the unit root in each of the series could be very different. For example, in terms of the common trend repre­sentation of a bivariate system discussed above, it could well be the case that y1t = ф1 yC + +1t and y2t = ф2yC + +2t are such that ф1 is close to zero and that ф2 is large. Then y1t will not be different from +1t which is an I(0) series while y2t will be clearly I(1). The two series are cointegrated, since they share a common trend. However, if we regress y1t on y2t, i. e. we normalize the cointegrating vector on the coefficient of y1t, the regression will be nearly unbalanced, namely, the regressand is almost I(0) whilst the regressor is I(1)...

The formulation of explicit statistical models for time series began with the classic papers of Yule (1927) (Autoregressive (AR( p)) scheme):

p

yt = ao + X akVt-k + Ef, Et ~ NI(0, о2), t = 1, 2,…,

k=1

where "NI" stands for "Normal, Independent" and Slutsky (1927) (Moving Average (MA(q)) scheme):

yt = Yo + X YkE-k + Et, Et ~ NI(0, о2), t = 1, 2,….

k=1

Viewing these formulations from today’s vantage point, it is clear that, at the time, they were proposed as nothing more than convenient descriptive models for time series data. Their justification was based exclusively on the fact that when simulated these schemes gave rise to data series which appear to exhibit cycles similar to those observed in actual time series data...

Recall that the exponential duration model depends on the parameter X, which is the constant hazard rate. We now assume an exponential distribution for each individual duration, with a rate Xi depending on the observable characteristics of this individual represented by explanatory variables. The positive sign of X is ensured by assuming that:

Xi = exp(xi0),

where 0 is a vector of unknown parameters. The survivor function is given by:

Si(y | xy 0) = exp[-(expx;0)y],

whereas the conditional pdf of the duration variable given the covariates is:

f (yi | x; 0) = X;exp(-Xiyi)

= exp(xi0)exp[-yi exp(x;0)]. (21.9)

The parameter 0 can be estimated by the maximum likelihood from a random sample of N observations on (x„ y;), i = 1,…, N. The conditional loglikelihood function is:

The Bayesian fixed effects model described above might initially appeal to re­searchers who do not want to make distributional assumptions about the ineffi­ciency distribution. However, as we have shown above, this model is implicitly making strong and possibly unreasonable prior assumptions. Furthermore, we can only calculate relative, as opposed to absolute, efficiencies. For these reasons, it is desirable to develop a model which makes an explicit distributional assump­tion for the inefficiencies. With such a model, absolute efficiencies can be calcu­lated in the spirit of the cross-sectional stochastic frontier model of Section 2, since the distribution assumed for the zi s allows us to separately identify zi and в 0...

At this point it is important to emphasize that the above discussion relating to the convergence of certain partial sums of the MA(«) coefficients is not helpfulfrom the empirical modeling viewpoint because the restrictions cannot be assessed a priori. Alternatively, one can consider restrictions on the temporal covariances of the observable process {yt, t Є T} which we can assess a priori:

1d

constant mean:

E(yt) := P, t Є T,

2d

constant variance:

var( yt) := a0, t Є T,

3d

ш-autocorrelation:

, , Кк т 1 ^ . . . , ^ cov(yu yt-т) := n

[0, t > q,

4d

normality:

yt ~ N(-,.), t Є T. (28.22)

where the first two moments in terms of the statistical parameterization ф := (a0, a1,…, aq, a2) take the form:

The characteristics of the variables involved determine to some extent which model is a suitable representation of the data generation process (DGP). For instance, the trending properties of the variables and their seasonal fluctuations are of importance in setting up a suitable model. In the following a variable is called integrated of order d (I(d)) if stochastic trends or unit roots can be removed by differencing the variable d times (see also Chapter 29 by Bierens in this vol­ume). In the present chapter it is assumed that all variables are at most I(1) if not otherwise stated so that, for any time series variable yit it is assumed that Ayit = yit – Vk, t-1 has no stochastic trend...